Problem 15 - Algebra 2

Question

ELECTRICITY The current in one part of a series circuit is \(4-j\) amps. The current in another part of the circuit is \(6+4 j\) amps. Add these complex numbers to find the total current in the circuit.

Step-by-Step Solution

Answer

The total current in the circuit is \(10 + 3j\) amps.

Step 1: Identifying Complex Numbers

We need to add the two complex numbers: \(4-j\) and \(6+4j\). Each part of these numbers consists of a real and an imaginary component.

Step 2: Separating Real and Imaginary Parts

The first complex number, \(4-j\), has a real part of \(4\) and an imaginary part of \(-1\). The second complex number, \(6+4j\), has a real part of \(6\) and an imaginary part of \(4\).

Step 3: Adding Real Parts

Add the real parts from both numbers: \(4 + 6 = 10\).

Step 4: Adding Imaginary Parts

Add the imaginary parts from both numbers: \(-1 + 4 = 3\).

Step 5: Forming the Resulting Complex Number

Combine the results from the previous steps. The total current expressed as a complex number is \(10 + 3j\).

Key Concepts

Series Circuit

A series circuit is a type of electrical circuit in which components are connected end-to-end in a single path. This means that the current, which carries electrical energy, flows through each component one by one. There are a few important characteristics of series circuits that make them unique:

• Single Pathway: The electrical current has only one path to follow. This implies that if one component fails or is removed, the entire circuit stops functioning.
• Current Consistency: The same current flows through all components because it has only one path to take. This is unlike parallel circuits where current may vary between branches.
• Total Resistance: In a series circuit, the total resistance is the sum of the resistances of all components. This affects how easily the current flows.

Understanding series circuits is crucial when analyzing complex numbers related to current, as we'll see when we add real and imaginary parts of complex numbers in a series circuit.

Real and Imaginary Parts

Complex numbers have two fundamental components: a real part and an imaginary part. These components give them unique properties, especially useful in electrical engineering.

• Real Part: This is the 'normal' number part of a complex number, such as the 4 in the complex number \(4-j\). It's represented without any 'j' or imaginary component and often describes real-world measurements.
• Imaginary Part: In electrical circuits, imaginary parts are used to represent quantities that, while not directly measurable, still impact the system. The imaginary part uses the imaginary unit \(j\), such as the \(-1\) in \(4-j\). Here, \(j\) represents \(\sqrt{-1}\).

By separating these two parts, calculations like adding or subtracting complex numbers become more manageable. Recognizing these components will make tasks like analyzing current in series circuits much easier.

Adding Complex Numbers

Adding complex numbers, especially in contexts like series circuits, involves a straightforward process that ensures calculations remain organized and clear.

1. Identify the real and imaginary parts: For example, in the complex numbers \(4-j\) and \(6+4j\), identify \(4\) and \(6\) as real parts, and \(-1\) and \(4\) as imaginary parts.
2. Add real parts together: Simply sum the real components. For the numbers \(4-j\) and \(6+4j\), the addition is \(4 + 6 = 10\).
3. Add imaginary parts together: Sum the imaginary components. For our example, this would be \(-1 + 4 = 3\).
4. Formulate the resulting complex number: Combine the summed real and imaginary parts to find the total, \(10+3j\), which in this context represents the total current in the circuit.

By following these steps, you can easily manage complex numbers, a critical skill in both academic exercises and real-world electrical engineering applications.